So let me preface this by saying that the concept of trans people sort of entered my consciousness through social media (mainly YouTube and Twitter), so I've known about them for a while now, but I've always had questions regarding certain things that I just didn't know where to ask till now, so I hope no one takes any offense to these questions and please understand that if I come accross as insensitive, then that comes purely from ignorance rather than malicious intent. So here they are:

1. First, let's start with the definition, if I'm understanding this right, A trans person is someone whose "sex assigned at birth does not match with their gender identity", in trans theory, it is said that sex and gender are two separate things(the former being a biological thing and the latter being a social construct), now if you go by the definition mentioned earlier, then doesn't it imply that a certain gender supposedly corresponds to a certain sex? If not, how is there a "mismatch" between them in a trans person? Or why do trans people take HRT/sex change surgery which changes the sex?
2. According to trans theory, a person could be born male(sex) but identify as a woman(gender), except they think that their gender and its attributes are more associated with the male sex rather than the female sex, in that case, is that person still trans? In fact, at that point, what do the terms "trans" or "cis" even mean anymore?
3. This is perhaps going to be the most controversial one, so apologies in case this offends someone and please bear with me, but am i wrong for thinking is?its just that sometimes.... I just can't help but feel like as if the whole total separation between gender and sex is just one massive cope to deal with the fact that current technology just isn't enough to fully transform someones sex, I'm sure that there are trans people who are saddened by the fact that they just can't do certain things, even sex change surgery doesn't magically grant you the ability to have children for example, I'm sure that there's a sizeable portion of trans women who would like to conceive children except that they just... can't. Maybe in the future, the technology will get there but currently its just not enough...

If you've made it this far, then thanks for putting up with me, my questions may seem dumb but that's becuas I genuinely don't know and am trying to understand.